itseadjointisia

Itseadjointisia (self-adjoint operators) are linear operators on a Hilbert space that are equal to their own adjoint. In a complex Hilbert space H, a densely defined operator T with domain D(T) is self-adjoint if T equals its adjoint T*, meaning D(T) = D(T*) and ⟨Tx, y⟩ = ⟨x, Ty⟩ for all x and y in D(T). Equivalently, T is self-adjoint when it is symmetric (⟨Tx, y⟩ = ⟨x, Ty⟩ for all x, y in D(T)) and has no proper self-adjoint extension.

In finite dimensions, self-adjoint operators coincide with Hermitian matrices, or real symmetric matrices when the underlying field is real. In infinite dimensions, self-adjointness requires careful attention to domains; many symmetric operators are not self-adjoint unless their domains are chosen appropriately. When a symmetric operator is not self-adjoint, it may admit self-adjoint extensions, and the existence and uniqueness of such extensions are described by deficiency indices.

A central feature of ise adjointisia is their spectral theory. Self-adjoint operators have real spectra, and there exists a spectral theorem that allows representing T as an integral with respect to a projection-valued measure. This enables a functional calculus: for a broad class of functions f, one can define f(T) in a consistent, stable way.

Examples include finite-dimensional Hermitian matrices, which are self-adjoint, and infinite-dimensional operators such as the Laplacian with appropriate boundary conditions, or the momentum operator on L^2 spaces with a suitable domain. Self-adjoint operators play a foundational role in quantum mechanics, where observables are modeled by self-adjoint operators, and in the study of differential equations and boundary-value problems.